The interplay between Seiberg-Witten theory and complex geometry has been very fruitful since the early development of the theory (see for example [1]},[2]},[3]} among the many). Following this thread, in this paper we investigate the interactions between monopole Floer homology [4]}, a package of topological invariants of three-manifolds equipped with a spin\(^c\)  structure \((Y,\mathfrak {s})\) , and the classical study of the geometry of algebraic curves.
